Finite Math Examples

$- \frac{x^{4} - x^{2} - 5}{x^{2} + 6} < 0$

Step 1

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x^{4} - x^{2} - 5 = 0$

$x^{2} + 6 = 0$

Step 2

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} - u - 5 = 0$

$u = x^{2}$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 1$ , $b = - 1$ , and $c = - 5$ into the quadratic formula and solve for $u$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot - 5)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 1$ to the power of $2$ .

$u = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{1 \pm \sqrt{1 - 4 \cdot - 5}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $- 5$ .

$u = \frac{1 \pm \sqrt{1 + 20}}{2 \cdot 1}$

Step 5.1.3

Add $1$ and $20$ .

$u = \frac{1 \pm \sqrt{21}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$u = \frac{1 \pm \sqrt{21}}{2}$

Step 6

The final answer is the combination of both solutions.

$u = \frac{1 + \sqrt{21}}{2}, \frac{1 - \sqrt{21}}{2}$

Step 7

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = 2.79128784$

${(x^{2})}^{1} = - 1.79128784$

Step 8

Solve the first equation for $x$ .

$x^{2} = 2.79128784$

Step 9

Solve the equation for $x$ .

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Step 9.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{2.79128784}$

Step 9.2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 9.2.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{2.79128784}$

Step 9.2.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{2.79128784}$

Step 9.2.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{2.79128784}, - \sqrt{2.79128784}$

Step 10

Solve the second equation for $x$ .

${(x^{2})}^{1} = - 1.79128784$

Step 11

Solve the equation for $x$ .

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Step 11.1

Remove parentheses.

$x^{2} = - 1.79128784$

Step 11.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 1.79128784}$

Step 11.3

Simplify $\pm \sqrt{- 1.79128784}$ .

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Step 11.3.1

Rewrite $- 1.79128784$ as $- 1 (1.79128784)$ .

$x = \pm \sqrt{- 1 (1.79128784)}$

Step 11.3.2

Rewrite $\sqrt{- 1 (1.79128784)}$ as $\sqrt{- 1} \cdot \sqrt{1.79128784}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{1.79128784}$

Step 11.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \sqrt{1.79128784}$

Step 11.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 11.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i \sqrt{1.79128784}$

Step 11.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i \sqrt{1.79128784}$

Step 11.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i \sqrt{1.79128784}, - i \sqrt{1.79128784}$

Step 12

The solution to $x^{4} - x^{2} - 5 = 0$ is $x = \sqrt{2.79128784}, - \sqrt{2.79128784}, i \sqrt{1.79128784}, - i \sqrt{1.79128784}$ .

$x = \sqrt{2.79128784}, - \sqrt{2.79128784}, i \sqrt{1.79128784}, - i \sqrt{1.79128784}$

Step 13

Subtract $6$ from both sides of the equation.

$x^{2} = - 6$

Step 14

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 6}$

Step 15

Simplify $\pm \sqrt{- 6}$ .

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Step 15.1

Rewrite $- 6$ as $- 1 (6)$ .

$x = \pm \sqrt{- 1 (6)}$

Step 15.2

Rewrite $\sqrt{- 1 (6)}$ as $\sqrt{- 1} \cdot \sqrt{6}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{6}$

Step 15.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \sqrt{6}$

Step 16

The complete solution is the result of both the positive and negative portions of the solution.

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Step 16.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i \sqrt{6}$

Step 16.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i \sqrt{6}$

Step 16.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i \sqrt{6}, - i \sqrt{6}$

Step 17

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = \sqrt{2.79128784}, - \sqrt{2.79128784}$

$x = i \sqrt{6}, - i \sqrt{6}$

Step 18

Consolidate the solutions.

$x = \sqrt{2.79128784}, - \sqrt{2.79128784}$

Step 19

Use each root to create test intervals.

$x < - \sqrt{2.79128784}$

$- \sqrt{2.79128784} < x < \sqrt{2.79128784}$

$x > \sqrt{2.79128784}$

Step 20

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 20.1

Test a value on the interval $x < - \sqrt{2.79128784}$ to see if it makes the inequality true.

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Step 20.1.1

Choose a value on the interval $x < - \sqrt{2.79128784}$ and see if this value makes the original inequality true.

$x = - 4$

Step 20.1.2

Replace $x$ with $- 4$ in the original inequality.

$- \frac{{(- 4)}^{4} - {(- 4)}^{2} - 5}{{(- 4)}^{2} + 6} < 0$

Step 20.1.3

The left side $- 10.6\overline{81}$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 20.2

Test a value on the interval $- \sqrt{2.79128784} < x < \sqrt{2.79128784}$ to see if it makes the inequality true.

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Step 20.2.1

Choose a value on the interval $- \sqrt{2.79128784} < x < \sqrt{2.79128784}$ and see if this value makes the original inequality true.

$x = 0$

Step 20.2.2

Replace $x$ with $0$ in the original inequality.

$- \frac{{(0)}^{4} - {(0)}^{2} - 5}{{(0)}^{2} + 6} < 0$

Step 20.2.3

The left side $0.8\overline{3}$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 20.3

Test a value on the interval $x > \sqrt{2.79128784}$ to see if it makes the inequality true.

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Step 20.3.1

Choose a value on the interval $x > \sqrt{2.79128784}$ and see if this value makes the original inequality true.

$x = 4$

Step 20.3.2

Replace $x$ with $4$ in the original inequality.

$- \frac{{(4)}^{4} - {(4)}^{2} - 5}{{(4)}^{2} + 6} < 0$

Step 20.3.3

The left side $- 10.6\overline{81}$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 20.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < - \sqrt{2.79128784}$ True

$- \sqrt{2.79128784} < x < \sqrt{2.79128784}$ False

$x > \sqrt{2.79128784}$ True

$x < - \sqrt{2.79128784}$ True

$- \sqrt{2.79128784} < x < \sqrt{2.79128784}$ False

$x > \sqrt{2.79128784}$ True

Step 21

The solution consists of all of the true intervals.

$x < - \sqrt{2.79128784}$ or $x > \sqrt{2.79128784}$

Step 22

The result can be shown in multiple forms.

Inequality Form:

$x < - \sqrt{2.79128784} or x > \sqrt{2.79128784}$

Interval Notation:

$(- \infty, - \sqrt{2.79128784}) \cup (\sqrt{2.79128784}, \infty)$

Step 23